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A review of volume changes in rubbers: the effect ofstretching

Jean-Benoit Le Cam

To cite this version:Jean-Benoit Le Cam. A review of volume changes in rubbers: the effect of stretching. RubberChemistry and Technology, American Chemical Society, 2010, 83, pp.247-269. �10.5254/1.3525684�.�hal-01131582�

1

A REVIEW OF VOLUME CHANGES IN RUBBERS: THE EFFECT OF

STRETCHING

JEAN-BENOÎT LE CAM

CLERMONT UNIVERSITÉ, INSTITUT FRANÇAIS DE MÉCANIQUE AVANCÉE, EA 3867,

LABORATOIRE DE MÉCANIQUE ET INGÉNIERIES, BP 10448, F-63000 CLERMONT-FERRAND

CONTENTS

Page

I. Introduction ........................................................................................................................ 1

II. The measurement of volume variation ............................................................................... 3

III. Experimental results ........................................................................................................... 3

A. The works of schippel (1920) and feuchter (1925) .................................................... 4

B. The work of Holt and McPherson (1936) .................................................................. 6

C. 1940-1980: most of the measurements are used for volume change modeling ......... 8

D. The most recent studies: toward the comprehension of deformation mechanisms .. 10

IV. Modeling volume change in stretched rubbers ................................................................ 14

A. Modeling the reversible change in volume under uniaxial loading conditions ........ 15

B. Modeling the reversible change in volume under multiaxial loading conditions .... 23

C. Note on modeling the irreversible change in volume under multiaxial loading

conditions ......................................................................................................................... 27

V. Conclusions and perspectives ........................................................................................... 28

VI. References ........................................................................................................................ 30

I. INTRODUCTION

Before the 20th century, only a few brief observations on rubber properties were

reported; for instance those of Gough1 (1805) and Joule

2 (1857), which dealt with the increase

in temperature of rubber when stretched and are classically cited in the literature. The first

studies that investigated the mechanical properties of rubber date from the beginning of the

20th century. A major result was the observation of the decrease in stiffness of rubber during

2

the first mechanical cycles by Bouasse and Carrière3 (1903). Later, this phenomenon was

studied more precisely by Holt4 (1931) and Mullins

5 (1948) and was then referred to as "the

Mullins effect". Other physical phenomena involved in the deformation of rubber were also

characterized by using advances in physics and technology. For example, x-ray diffraction

contributed to the comprehension of the kinetics of polymer chain crystallization. Today,

rubber is still the object of vivid scientific interest and a material with "extraordinary physical

properties".6

To investigate these physical properties, several mechanical quantities have been used.

Among them, the change in volume of stretched rubber seems to be one of the most relevant,

in particular to analyze the change in the rubber microstructure. The first record of this

phenomenon dates back as far as the end of the nineteenth century in the works of Joule6

(1884). The author observed that the specific gravity of natural rubber decreased upon

stretching it (about 0.15 per cent for a 100 per cent stretch). Even though these results were

corroborated by Mallock7 (1889) while investigating the physical properties of vulcanized

India-Rubber, Thomson8 stated (1890) that a column of rubber when stretched out suffers no

significant change in volume and rubber may therefore be regarded as an incompressible

elastic solid. Then, while studying the nature of the stress-strain curves for rubber containing

different pigments in varying quantities, Schippel9 (1920) considered that possibly, when the

rubber was sufficiently stretched, it might pull away from the particles of pigment along their

axes of stress and cause vacua to be formed on either side of each particle, and a considerable

increase in the rubber body volume might therefore be observed. Later, the discovery of the

ability of rubber chains to crystallize under stretching or at low temperature motivated a

number of authors to investigated more precisely the change in volume of rubbers, under

stretching and/or under heating and cooling.

3

The aim of the present paper is to give an exhaustive view of the state of knowledge

on volume variation and to link it to the physical phenomena observed. For this purpose, the

results obtained experimentally are first presented in a chronological manner. Special

attention is paid to detailing their relative contribution. Secondly, the different approaches

adopted to modeling the isothermal volume changes in stretched rubbers are summarized.

Finally, conclusions and perspectives close the paper.

II. THE MEASUREMENT OF VOLUME VARIATION

Numerous measurement methods have been used to characterize the change in volume

of stretched rubbers. They used various technologies whose accuracy differed from one to

another. Moreover, loading and temperature conditions are not precisely described. This

explains why the results obtained are not easily comparable. The technology classically used

to measure volume variation is dilatometry. In this case, the measurement of the change in

volume of rubber is performed by enclosing samples in suitable confining liquids in

dilatometers and observing the changes in the height of the liquid in calibrated capillary tubes

when the rubber is stretched. Smaller capilaries give more precise values.4,10,11

Certain authors

developed their own procedure to measure the volume change, which can be determined by

calculating the change in specific gravity9, measuring the hydrostatic weighting

12-14 or

measuring the relative pressure using the Farris principle15

. Optical measurement methods

have also been developed recently by Le Cam and Toussaint.16,17

III. EXPERIMENTAL RESULTS

4

As previously indicated in the introductory section, even though preliminary

observations of the change in volume of stretched rubber were briefly reported by

investigators at the end of the 19th

century6-8

, the first quantitative studies date from the

beginning of the 20th century. In this section, we propose to review the main results obtained

from this period in a chronological manner.

A. THE WORKS OF SCHIPPEL (1920) AND FEUCHTER (1925)

The work of Schippel9 was the first to investigate quantitatively the change in volume

of stretched natural rubber. Schippel investigated the volume change in stretched rubber

containing various pigments. This study was motivated by previous experiments by the author

on transparent vulcanized compounds containing a fair proportion of medium-sized lead shot.

When the compound was stretched, the formation of vacua proceeded gradually until each

lead shot had its own conical vacuum on either side in the direction of strain. To investigate

more precisely this phenomenon, this author added pigments (here given in phr (part per cent

of rubber) by volume) in varying quantities to the rubber matrix: barytes, i.e. barium sulfate

(BaSO4), from 0 to 150, whiting from 0 to 150, zinc oxide from 0 to 125, china clay from 0 to

25, red oxide from 0 to 30, lamp carbon from 0 to 75 and carbon black from 0 to 30. The tests

were carried out until sample fracture. For each particle type, the increase in volume was

measured every 50 per cent strain increase up to a 200 or 300 per cent strain. The final curve

describing relative volume variation versus elongation was obtained by linking the last point

at 200 or 300 per cent strain to the point corresponding to the sample fracture. This is

illustrated by the diagram in Figure 1. Results obtained by Schippel showed that the higher

the quantity of particles, the higher the volume increase and the lower the elongation before

fracture. The maximum of change in volume was found to attain 120% for the barite particles.

5

Regarding the effect of zinc oxide, Schippel highlighted that a threshold of 40 phr in volume

was necessary to obtain a higher increase in volume for a given elongation, and concluded

that below this threshold, zinc oxide had no effect on volume change. The author also found

that the mean diameter of the particles had the same effect as the quantity. Finally, Schippel

chose the level of change in volume as a relevant measure of adhesion and classified red

oxide, zinc oxide, lamp black and carbon black as particles that exhibited a strong adhesion

with the rubber matrix, contrary to barites and whiting. In Figure 1, this latter observation can

be illustrated by angle α, which is large for barites and whiting, and small for the other

particles. The fact that the volume can not exceed a certain value, modeled by a value of angle

α close to π/2 rad, seems to be a criterion of high adhesion between particles and the rubber

matrix for Schippel.

Later, a brief study by Feuchter18

reported some results that differed from those of

Schippel by showing that the volume of natural rubber decreased with elongation. The author

explained this result by the formation of an anisotropic system in stretched natural rubber,

namely fibering or crystallization. This result, which seems to be contradictory to that of

Schippel, is the first that envisaged that volume might decrease through the contribution of

phenomena that tend to reorganize the polymer chains. Moreover, compared to the previous

work of Schippel, the fact that no increase

in volume of rubber was observed can be explained by the fact that Feuchter studied an

unfilled natural rubber. Thus, the decrease in volume of rubber due to crystallization is a first-

order phenomenon compared to the cavitation phenomenon. In Schippel's study, the presence

of fillers was favorable and amplified the cavitation. Finally, the decrease in the volume of

rubber was observed by Feuchter beyond a certain elongation threshold in the range of

elongation for which Schippel had not measured the volume. Even though a second-order

phenomenon had taken place, Schippel did therefore not detect it.

6

At this stage of the present literature survey, the volume variation in rubber seems to

be affected by two major phenomena: nucleation and the growth of vacua around the

particles, which leads to an increase in volume, and the reorganization of polymer chains

beyond a certain elongation threshold, namely crystallization, which leads to the opposite

effect. With regard to these phenomena, some questions of importance arise: Are they

reversible? Are they simultaneous? Is the elongation at which crystallization occurs constant

or is it dependent on time, polymer chains, polymer network and/or temperature? The first

observations previously summarized illustrate the complexity of the phenomena that occur

during the deformation of rubber. This is the reason why a number of experimental techniques

were used to investigate this deformation. For instance, by means of x-rays, a time lag was

observed in the double refraction of stretched rubber.19-21

This time lag is also necessary to

obtain the stabilization of the volume at a given elongation22

. Again, this could explain the

phenomenon called "optical creep" and observed by photoelasticity by Thibodeau and

McPherson.23

These results motivated the study on the change in volume of stretched rubber

by Holt and McPherson11

.

B. THE WORK OF HOLT AND MCPHERSON (1936)

In this work11

, the authors conducted two series of experiments with unfilled samples

vulcanized with sulfur. In the first series, the rubber was stretched at 25°C to a given

elongation and held at that elongation for 3 minutes, observations on the volume being made

at frequent intervals. It was then released and observations were again taken at intervals over

a period of 3 minutes. The same procedure was repeated for various successive elongations.

The results are illustrated by the diagram in Figure 2. A decrease in volume corresponding to

0.1% in 3 minutes was observed beyond an elongation equal to 450%. At higher elongations

7

the decrease became progressively greater and amounted to nearly 2% at an elongation of

725%.

In a second series of experiments the samples were stretched at 25°C first to a

relatively low elongation of 200 percent, and were held at this elongation for 3 minutes while

observations were made. Then, without release, the sample was stretched to successively

higher elongations, each for 3 minutes, until the maximum elongation had been reached. The

process was then reversed by the same stepwise procedure. The results are illustrated in

Figure 3. The changes in volume at different elongations were slightly greater than the

changes which were observed when the samples were released between successive

elongations. Moreover, when the sample was released in a stepwise manner the volume

showed a definite lag, and at any elongation except zero it was less than the volume attained

on stretching to the same elongation. At zero elongation, the original volume was recovered.

The authors interpreted these results as the fact that the release phase might correspond to the

equilibrium state, but they reported that this was not in agreement with the fact that when

rubber is held for a longer time in the stretched condition, its volume may reach lower values.

The authors also explained that there is an elongation threshold of about 450% beyond which

volume variation became apparent. Thus, the results of Holt et al. were the first from which

the elongation at the beginning of crystallization can be deduced. Indeed, during stretching,

the decrease in volume became apparent when the elongation at the beginning of

crystallization was reached. Moreover, the time necessary to stabilize the volume at a given

elongation was higher during stretching than during the release phase. The authors concluded

that the change in volume during stretching is influenced by the same considerations as X-ray

diffraction, since it is observed only above a certain critical elongation and is greater the

higher the elongation, the lower the temperature, and the longer the time the sample is

maintained stretched.

8

Finally, this study by Holt and McPherson gave the first observations on the influence

of repeated cycles on the change in volume: after a first set of elongations was applied and the

samples were maintained stretched, the same set of elongations was applied again to the

samples. The authors concluded that the change in volume on stretching was not significantly

affected by the previous stress history of the samples and this is therefore in contrast to stress-

strain behavior, which is markedly altered by the first few stretching cycles.3-5

C. 1940-1980: MOST OF THE MEASUREMENTS ARE USED FOR VOLUME

CHANGE MODELING

During this period, a number of authors tended to predict the volume change in rubber

from measurements at low elongations.12,24-29

Some of these authors cast doubts on the

measurements performed by Holt and MacPherson for low elongation values (up to 200%

elongation). For instance, Gee12

considered that the dilatometric method used by Holt and

McPherson was not sufficiently sensitive to measure volume change in this elongation range,

and performed the measurement with a new apparatus using hydrostatic weighting; they

found, contrary to Holt and McPherson, a significant volume variation of about 2 10-4

at

125% elongation. In the larger elongation ranges, the studies that attempted to model the

volume change at elongations superior to 200% were conducted with filled synthetic rubbers

such as styrene butadiene rubber30

and Viton31

, i.e. non-crystallizable rubbers for which the

volume increases in a monotonic manner when stretched. The models proposed in the

literature are summarized in the next section.

It should be noted that, in this period, the work of Mullins and Tobin13

is the only one

that investigated volume variation in stretched rubber, unfilled or filled by carbon black,

vulcanized using sulphur or peroxides, in the large elongation ranges. This work reconciled

the previous results on volume variation at low elongations with those obtained by Holt and

9

McPherson at large elongations: Mullins and Tobin showed that the volume always increased

before decreasing. The decrease is the consequence of crystallization, which takes place

above a certain elongation value. For filled rubber vulcanized with sulphur, the decrease

began between 100 and 200% elongation. For unfilled rubber vulcanized with sulphur, the

maximum applied elongation of 400% was not sufficient to highlight a decrease in rubber

volume. This seems to indicate that fillers act, at the microscopic scale, as concentrators of the

deformation and allow crystallization to occur at lower macroscopic elongation. For unfilled

rubber vulcanized using peroxides, the decrease began between 200 and 300% elongation

(this result was also found by Reichert, Hopfenmüller and Göritz.32

Compared to

vulcanization with sulphur, this observation seems to indicate that vulcanization with

peroxides leads to a higher deformation localization than vulcanization with sulphur and thus

acts like fillers do. Finally, Shinomura and Takahashi33

measured the volume changes in

carbon black-filled butyl and styrene butadiene rubbers and proposed to distinguish two parts

in the response of the materials in terms of relative volume variation. As shown in Figure 4,

volume variation versus elongation can be modeled by two curves, each of them

corresponding to one type of cavitation. The authors explained that the first type of cavitation

originates in the breakdown of carbon black-rubber interactions and the second type comes

from the breakdown of carbon black aggregates. Thus, they proposed to analyze volume

variation by introducing a new mechanical quantity defined as the ratio between the volume

variation and the uniaxial Cauchy stress σ. Figure 5 shows the results obtained. As previously,

in Figure 4, this relation is modeled by two curves, corresponding to the first and second type

of cavitation. The first curve exhibited a plateau whose level depends on the filler structure

and the interaction between fillers and the rubber matrix.

10

Finally, elongation threshold is observed before volume variation. It should be noted

that in this work the uniaxial Cauchy stress is calculated using the assumption that the

material is incompressible.

D. THE MOST RECENT STUDIES: TOWARD THE COMPREHENSION OF

DEFORMATION MECHANISMS

In the recent past, fives studies have been dedicated to volume variation in stretched

rubbers. They were carried out on synthetic and natural rubbers and tended to link the

measurements to the mechanisms of volume change and consequently of deformation. The

first two investigated the nature and the surface treatment of fillers in synthetic rubber. The

study by Kumar et al.14

(2007) dealt with the incorporation of recycled rubber granulates,

considered as intrinsic flaws, in a virgin styrene butadiene rubber matrix. No volume change

with strain was observed in this matrix, unfilled or filled with 70 phr carbon black

aggregates (N330, HAF). Authors measured the volume variation in stretched compounds

with various granulate size and modulus and investigated the change in flaw size with strain

and the reduction in strength resulting from a weaker interface using a microstructural finite

element analysis. Results highlighted that flaw size increases in a characteristic way with

strain if the rubber matrix and granulates have a similar modulus, whereas a modulus

mismatch results in much larger volume changes and hence greater flaw size which also

appears to increase with strain. As a perspective, authors suggested that their approach would

be well-suitable to evaluate the effectiveness of surface modification techniques in the

future.34-36

This was the aim of the study of Ramier et al.37

(2007) that focused on the influence of

the treatment of silica fillers in styrene butadiene rubber. These fillers were treated with either

a covering agent or a coupling agent. First, tests performed previously on the rubber matrix

11

had not highlighted any change in volume. Second, without any treatment, silica-filled styrene

butadiene rubber exhibited the highest volume variation at each applied elongation. Third, for

both silica filler treatments, volume variation increased monotonically with elongation. Using

the covering agent, the higher its quantity, the lower the volume variation. Using the coupling

agent, the curve is the same whatever its quantity. Moreover, as illustrated in Figure 6, the

concavity of the curves obtained for the two treatments was different. By plotting volume

variation as a function of stress, the authors concluded that the covering agent is favorable to

decohesion and void formation phenomena and that the coupling agent delays the occurrence

of decohesion because of the strong cohesion between treated silica fillers and the rubber

matrix.

After these first results on styrene butadiene rubber, some of the previous authors

attempted to determine parameters governing strain-induced crystallization in filled natural

rubber.15

For this purpose, the authors measured simultaneously the tensile stress, the volume

variation and the crystallinity of filled and unfilled natural rubbers (Standard Malaysian

Rubber number 10). Three kinds of carbon black (N324, N347 and N330) and one type of

silica were used to fill the samples (45 phr in weight for carbon black and 50 for silica). Tests

were performed over several cycles at a temperature and a strain rate set at 20°C and 0.25

min-1

, respectively. Figure 7 illustrates the results obtained with all the samples. During

stretching, filled natural rubbers exhibited a positive volume variation due to filler-rubber

decohesion and cavitation in the rubber matrix. The maximum value of the positive volume

variation (less than 4% for carbon black filler referred to as N234) decreased with each cycle

until stabilization. Even through the authors located the stabilization of the tensile stress after

the third cycle, they did not discuss the number of cycles necessary to stabilize the volume

variation. During unloading, they observed that for a given elongation the volume variation is

lower and they explained this result by a fast recovery of decohesion /cavitation. They also

12

observed that volume variation became negative before returning to zero. For the authors, this

is the signature of strain-induced crystallization, which tends to reduce significantly the

volume of rubber. To highlight the influence of fillers, they measured the volume change for

the same natural rubber, unfilled. It should be noted that they did not indicate whether the

curve was that of the stabilized cycle or not. For unfilled natural rubber, the maximum value

for volume variation reached 2% and no negative value was observed. This could be

explained by the fact that the elongation at the beginning of crystallization had not been

attained during stretching. They proposed to illustrate the contribution of both

cavitation/decohesion and crystallization phenomena to the global change in volume

measured by the diagram in Figure 8. The authors explained that they were not able to analyze

more precisely their results in terms of change in volume between the different filled samples

because the accuracy of the pressure sensor they chose for the apparatus used to measure the

volume was not high enough.

At this stage in the present state-of-the-art review, it seems relevant to be able to

distinguish the contribution of decohesion/cavitation and crystallization phenomena and to

study quantitatively the competition between the two phenomena through the measurement of

the change in volume. For this purpose, it is necessary to measure the volume variation more

accurately. This was the aim of the two last studies who introduced an original volume

measurement method.16-17

This is based on an optical measurement technique, namely digital

image correlation (D.I.C). In their work, Le Cam and Toussaint investigated the competition

between cavitation/decohesion and crystallization by detecting elongations at the beginning of

crystallization and at the melting of crystallites.16

Tests were performed on both natural and

synthetic rubbers, unfilled and filled with carbon black, and at a temperature, hygrometry and

strain rate set at 23°C, 34% and 1.3 min-1

, respectively. The results obtained are presented in

Figures 9 to 13. Figure 9 shows the relative volume variation obtained during the first

13

mechanical cycle for unfilled natural rubber. The authors modeled the curve in four segments

([OA], [AB], [BC] and [CD]) and described the competition between cavitation and stress-

induced crystallization in relation to each segment:

(i) segment [OA]: the volume increases due to the occurrence and growth of cavities in the

rubber matrix;

(ii) segment [AB]: from λA = 4.2, the volume begins to decrease. According to the authors,

even though cavities continue to appear and grow, the crystallization of the polymer chains

begins and is of a first-order phenomenon compared to cavitation. Consequently, in the

unfilled natural rubber considered in this study, the volume decreases;

(iii) segment [BC]: during unloading, the sample volume at a given stretch ratio is smaller

than during loading. According to the authors, this could be due to either the difference

between the kinetics of crystallization and of crystallite melting or the anelastic deformation

of cavities. To investigate the deformation of cavities, the authors measured the volume over

one cycle for which the maximum stretch ratio is still inferior to λA, i.e. the stretch ratio at

which crystallization is initiated. In terms of stress, the authors explained that the hysteresis

loop was not significant because no crystallization occurs in the bulk material. This result

corresponds to that of Trabelsi, Albouy and Rault38

. Figure 10 shows that the volume change

is the same for loading and unloading. The authors concluded that cavitation generated under

such loading conditions can be considered as an elastic process and that the kinetics of the

nucleation and growth of cavities and recovery can be considered as being the same on the

macroscopic scale. These results would indicate that the hysteresis loop obtained for volume

change curves is only due to chain crystallization. Finally, point C corresponds to the melting

of the last crystallites;

(iv) segment [CD]: the volume slightly decreases when the cavities close.

14

The authors then conducted the same measurements with the same natural rubber

filled with 34 phr of carbon black referred to as N326. Figure 11 presents the result obtained

in terms of relative volume variation for the first mechanical cycle. As shown in this figure,

the addition of fillers increased the volume variation. The fact that from λA = 1.64 the volume

variation does not decrease as in natural rubber indicates that, even though the elongation at

crystallization is lower than in natural rubber, the addition of fillers tends to amplify the

cavitation phenomenon and to minimize the level of crystallinity for a given stretch ratio. For

the authors, this is the reason why the hysteresis loop was smaller in filled than in unfilled

natural rubber. Moreover, the addition of fillers amplifies the local deformation and

consequently decreases the elongation at which crystallization begins.

To conclude their work on volume change in rubbers, Le Cam and Toussaint

performed cyclic tensile tests on natural and synthetic filled rubbers.17

Figures 12 and 13

show the third cycle in terms of relative volume variation obtained for filled natural rubber

and styrene butadiene rubber, respectively. For filled natural rubber, the hysteresis loop of the

relative volume variation curve obtained for the third mechanical cycle was lower than for the

first cycle. Moreover, the same characteristic stretch ratios as those of the first cycle were

observed: crystallization started at λ = 1.64 and the last crystallites melted at λ = 1.44. For the

authors, this seems to indicate that the Mullins effect has no influence on these characteristic

elongations. For filled styrene butadiene rubber, the first cycle was the only one that exhibited

a hysteresis loop in the volume change curve. No hysteresis loop was observed for the second

and third cycles and the evolution of volume variation versus stretch ratio was linear.

Moreover, no significant residual volume variation was observed.

These studies close the first part of the present review.

IV. MODELING VOLUME CHANGE IN STRETCHED RUBBERS

15

As highlighted above, the response of stretched rubbers in terms of volume change

comes from several physical phenomena. Obviously, the prediction of the relative volume

change of stretched rubber does not account for all of these phenomena and their modeling is

still a question of importance. This section presents the approaches proposed to model the

volume change. The first models date from the 1950s and predict the reversible change in

volume of isotropic rubber submitted to monotonic uniaxial stretching. Later, the prediction

of change in volume was performed for multiaxial loading conditions. After reporting the

uniaxial and multiaxial predictions of the change in volume of stretched rubber, a note is

dedicated to volume changes due to the irreversible process of deformation. It should be noted

that the influence of temperature on volume change is not discussed here.

A. MODELING THE REVERSIBLE CHANGE IN VOLUME UNDER

UNIAXIAL LOADING CONDITIONS

The first study that tried to model the change in volume of stretched rubber was that of

Gee25

(1946). The author assumed that the magnitude of the volume change can be estimated,

at least approximately, from the known dependence of the internal energy E of rubber on

isotropic changes in volume and used the well-known expression39

:

K

T

V

E

T

3

(1)

where TP

K

Vln is the isothermal compressibility and β is the coefficient of linear

expansion of the unstretched rubber. It should be noted that this equation does not take into

account the change in entropy due to phenomena such as crystallization of polymer chains

and the fact that β depends on the value of the elongation40

. Assuming that the isothermal

16

change of E with V is the same whether a hydrostatic pressure or a uniaxial tension is applied,

the volume variation obtained by stretching the rubber to length l is given by:

dll

El

T

KV

TP

l

l ,03

(2)

where l0 is the unstrained length of the rubber at temperature T . This equation applies as long

as the material remains isotropic. According to Gee25

, this assumption is valid for elongations

inferior to 2 in natural rubber but he did not discuss the frontier between isotropy and

anisotropy in terms of elongation. Moreover, Gee, examining the results of Holt and

McPherson11

in this range of elongation, assumed that the volume does not increase

significantly with the increase in temperature at constant pressure and deformation.

Considering his experimental data, the author rewrote Equation 2 as follows:

dll

flKV

l

l TP

0,3

1 (3)

where f is the uniaxial tensile force. Finally, Gee discussed the origin of volume and energy

changes and explained that for the classical theory of small elastic deformations, the applied

force may be resolved into shear stresses, which change the shape of the material without

affecting its volume, and a hydrostatic component, which changes the volume but not the

shape. The hydrostatic component induced by a unidirectional force is a tension p , equal in

magnitude to one-third of the tensile stress:

V

flp

3

1 (4)

and the resulting volume variation ΔV would equate pKV and consequently Kfl3

1. The

fact that this result can be obtained with the limiting form of Equation 3 as 0ll shows that,

for small deformations, the observed volume variation is produced by the hydrostatic

component of the tensile force. According to Gee, this consists of an increase in the average

17

intermolecular spacing, and is accompanied by equivalent increases in both internal energy

and entropy.

Later, Hewitt and Anthony26

(1959) rewrote Equation 3 in the form:

dBV

V

TP ,10 3

1

(5)

where B is the bulk modulus of the rubber, Π is the uniaxial component of the first Piola-

Kirchhoff tensor, P is the pressure of the surrounding fluid (about 1 atm) and λ is the ratio

between the deformed and undeformed lengths of the sample. This equation obviously applies

in the same elongation range as Equation 2, i.e. inferior to 2.

To predict the relative volume variation using Equation 5, it is necessary to estimate

Π. Here, we briefly recall the two theories classically applied. The first is the elementary

statistical theory for idealized networks and is based on the postulate that the elastic free

energy of a network is equal to the sum of the elastic free energies of the individual chains.41-

42 This leads us to neglect the intermolecular contributions to the total elastic free energy.

43-44

The expression of the elastic free energy is given by:

)3( 2

3

2

2

2

1 kTAel (6)

where the λi terms are the macroscopic principal extension ratios, i.e. the ratio between the

deformed and undeformed macroscopic dimensions of a prismatic test sample if the

macroscopic state of deformation may be assumed to be homogeneous. k is the Boltzmann

constant, T is the absolute temperature and depends on the model considered. It is equal to

2

for the affine network model,

45 where ν is the number of network chains. It is equal to

2

for the phantom network model,46

where

21 and is the average functionality, i.e.

the number of sites from which chains can grow. According to Flory, the elA expression

contains an additional logarithmic term that is a gas-like contribution resulting from the

18

distribution of the cross-links over the sample volume. Therefore, the affine model proposed

by Flory45

(1953) is rewritten as follows:

0

2

3

2

2

2

1 ln32 V

VkT

kTAel

(7)

where μ equals

2 , V is the final volume of the network, and V0 is the volume of network in

the state of formation.

The stress derives from the elastic free energy, according to the thermodynamic expression:47

VTi

elii

AV

,

1

(8)

where i is the Cauchy stress along the ith coordinate direction, i.e. yx

, and z

axes. The

subscripts T and V indicate that the differentiation is performed at fixed temperature and

volume.

The volume ratio is defined by:

321

0

V

V (9)

By considering that the volume of the network V without any applied force at the beginning of

the experiment may be different from V0, depending on the amount of solvent present relative

to that during formation, the final elongation is defined by:

iiV

V

3

1

0

(10)

where αi is the ratio between the final length of the network li and its initial undistorted length

at volume V with solvent Sil along the ith coordinate direction. In order words, this is similar

to the multiplicative decomposition of the deformation tensor gradient when considering the

19

continuum mechanics approach. The deformation comes from a purely dilatational part

(which is the swollen step here, before any stretching), and an isochoric part.

Equation 9 leads us to rewrite Equation 8 as:

i

S

S S

elii

AV

23

12

1 (11)

and using Equation 6:

3

1

2

S i

Sii

V

kT

(12)

In the following, we present the prediction of uniaxial volume change. It is therefore

necessary to develop the uniaxial stress-strain response. The deformation state along the x

axis for stretching and compression is given by:

3

1

0

1

V

V (13)

3

1

0

2

1

32

V

V (14)

where corresponds to 1 , i.e. the ratio of the final length along the direction of stretch to

the initial undistorted length at volume V. Thus, Equation 12 has the form:

1

3

1

2

11

S

S

V

kT (15)

and using Equation 14 leads to:

123

2

0

1 2

V

V

V

kT (16)

20

Finally, the retraction force f acting along the x

axis is obtained by multiplying both sides of

the previous equation by the deformed area:

23

2

01

2

V

V

L

kTf

i

(17)

where 1iL is the dimension of the network in the x

axis at the beginning of the experiment,

when the volume differs from V0 and depends on the quantity of solvent (see Figure 6.1, page

42 in reference 42). It should be noted that the retraction force can also be normalized by the

unstrained area. Thus, the stress-strain response of the material is given for low elongation

and at constant temperature by the statistical theory:

2

1

UT (18)

where U is related to the chain molecular weight Mc, the density ρ of the rubber, and the gas

constant R by the relation cM

RU

. depends on the particular version of the statistical

theory employed and on the structure of rubber.46

Its value is in the range 2

1 to 2.

Another one-parameter expression for Π was proposed by considering that Π remains

strictly proportional to strain for strains up to 10048

or 200%.49

Hence, the stress-strain

response is given by:

E (19)

where E is the Young's modulus of the material and ε the uniaxial linearized deformation,

defined by 0

0

l

ll .

An additional one-parameter stress-strain relationship was proposed

by Valanis and Landel50

(1967) in the logarithmic form:

21

2

3

2

11ln

3

2

E (20)

Finally for this first theory, a two-parameter relationship was proposed by Martin, Roth and

Stiehler51

(1956) which has the form:

1exp

112

AE (21)

where A is a parameter which depends on both the degree of crosslinking and the timescale.

The second theory used to model the stress-strain response of rubber-like material is

based on hyperelasticity, by considering the medium as a continuum. A number of models of

various complexity have been proposed; see for example: Mooney52

(1940); Treloar53

(1943);

Rivlin54,55

(1948); Biderman56

(1958); Hart-Smith57

(1966); Yeoh58,59

(1993, 1997);

Gent60

(1996); Haines and Wilson61

(1979); Gent and Thomas62

(1958); and Ogden63

(1972).

In the case of Mooney hyperelasticity:

2

12

12

CC (22)

where C1 and C2 are the material parameters of Mooney's law.

Using the expression of relative volume variation given by Equation 3, Fedors and

Landel30

(1970) proposed to compared the results obtained with the previous stress-strain

expressions. Thus, the calculation of the integral of Equation 3, using Equation 18, leads to

the expression of relative volume change:

12

2

1

3

2

0 B

UT

V

V (23)

This equation can be rewritten as follow26

:

22

12

2

1

9

2

0 B

E

V

V (24)

Using Equation 19, Equation 3 becomes:

ln30 B

E

V

V

(25)

Using Equation 20, Equation 3 becomes:

3

2

ln3

2

9

2

2

1

2

1

0

B

E

V

V (26)

These expressions of relative volume variation result from a one-parameter stress-strain

relationship. For instance, using Equation 22, the two-parameter expression of relative

volume variation would be:

1

2

1

221

0

12

3

2

32

2

1

3

2

C

C

C

C

B

C

V

V

(27)

and using the expression of Martin, Roth and Stiehler51

(1956):

dAAAAA

B

E

V

V

1

4

23

0

1exp

21

3 (28)

Once the value of A has been determined, the integral can be evaluated numerically. It should

be noted that Khasanovich64

(1959) pointed out that the material does not remain isotropic at

large deformations, and hence the integrand in Equation 5 should be multiplied by a factor μ

which takes into account the anisotropy of linear compressibility for a stretched material. If

the elastomer obeys the kinetic theory stress-strain law derived by James and Guth46

(1947),

then Khasanovich showed that:

2

32

(29)

and consequently Equation 5 becomes:

23

11

30 B

E

V

V (30)

The comparison carried out by Fedors and Landel30

highlighted that at small elongations, i.e.

less than 2, the predictions of the various proposals correspond well to the experimental data.

At large elongations, however, the predictions diverge. This is explained by the fact that Gee's

expression is formulated for small elongations and that for large deformations, other

phenomena such as crystallization and cavity nucleation and growth occur in the bulk

material.

B. MODELING THE REVERSIBLE CHANGE IN VOLUME UNDER

MULTIAXIAL LOADING CONDITIONS

To model the volume changes accompanying the deformation of rubber, the basis of

isotropic elastic theory is applied. We start with the multiplicative decomposition of the

deformation gradient ),( tXGradF

of a material point X

at time t into a volume-

changing part and a volume-preserving part:

FFF

(31)

),( tXx

denotes the deformation. The volume-preserving part is written:

1det,3

1

FFJF (32)

FJ det and the volume-changing part IJF 3

1

are used to define the unimodular left and

right Cauchy-Green tensors, respectively:

TT FFBFFC , (33)

1detdet BC , which can be expressed relative to the original Cauchy-Green tensors

FFC T and TFFB via:

24

BJBandCJC 3

2

3

2

(34)

The three first invariants of each of the previous tensors are defined by:

CIIItrCtrCIItrCI CCC det,2

1, 22

(35)

and

1,2

1, 222

JIIICtrCtrIICtrICCC

(36)

Based on the multiplicative decomposition defined in Equation 31, we assume a decoupled

constitutive representation of the free energy function:

CCCC

IIIWJUIIIJW ,,, (37)

so that the resulting stress state decomposes into a pure hydrostatic and a pure deviatoric part.

The volumetric part of the strain energy function U(J) is defined by JUK

and corresponds

to a penalty function. JU

denotes the principal function of the determinant J. One of the

simplest expression commonly used for JU

is 2)1(

2

1J . In the literature, many other

expressions of JU

have been proposed.63,65-72

The form of JU

is chosen with respect to

the convexity requirement for the volumetric part of the strain-energy function that implies

JU for 0J and J as well as 0'' JU so that a volumetric compression or

stretch yields hydrostatic pressure or tension72

.

Next, we present the prediction of relative volume variation expressed relative to the

principal stretches instead of the invariants of the Cauchy-Green tensors (see Equation 37).

For this purpose, the modified principal stretches i are introduced in the form

73:

3

1

Jii (38)

25

This expression is similar to Equation 10 where i corresponds to i , the ratio between the

final length of the network li and its the initial undistorted length at volume V with solvent Sil

along the ith coordinate direction. Thus:

1321 (39)

Thus in turn, the strain energy density of an isotropic elastic solid can be considered as a

function of 1 ,

2 , 3 and J, symmetrical in

1 , 2 ,

3 . The principal Cauchy stresses i

derive from the strain energy density according to:

3,2,1

i

WJ

i

ii

(40)

and using the modified principal stretches, Equation 40 becomes:

J

WJ

WJ

ji

j

j

ii

3

1

(41)

which can be rewritten as follows:

J

WJ

WWWJ

k

k

j

j

i

ii

3

1

3

1

3

2 (42)

With respect to uniaxial tension and considering that the relative volume change J-1 is of the

order of 0.01%, Ogden73

(1976) introduced the variable 1 J and the notation

JWJW ,,,, 2

1

2

1

and expanded JW ,

about J=1 to the

second power in ε. Thus, one can write:

1,2

11,1,,

2

22

J

W

J

WWJW

(43)

and the relative volume change ε is given by:

26

1,/1,1,2

12

2

J

WWW

(44)

Let us consider, for example, a certain class of strain energy densities introduced by

Ogden63

(1972):

JKgJWnnnn

n

n

3

1

3213 (45)

where the n s and n s are the material constants. The function g(J) is such that g(1) = g'(1) =

0, g''(1) = 1. Moreover, g'(J) > 0 if J > 1 and g'(J) < 0 if J < 1. The prime denotes

differentiation with respect to J. The principal Cauchy stresses associated with Equation 45

are:

JKJgJJn

n

ini '1 3

1

(46)

The author showed that Equation 44 can be approximated by:

22

1

1

n

K

n (47)

where K

. This expression corresponded to the work of Chadwick

74 (1974) and

Flory47

(1961), applied only to small stretches.

For higher stretches, the volume reaches a maximum, then falls to its initial value

before decreasing steadily until rupture. To account for such a phenomenon, Ogden proposed

to add a term, which involves a coupling between J and the modified stretches, to the strain-

energy function. Thus, for uniaxial tension, the relative volume change is given by:

n

nK 2

1

1 (48)

for equi-biaxial tension:

n

nK2

1 (49)

27

and for pure shear:

n

nK 1 (50)

It should be noted that Sharda and Tschoegl75

(1976) used the same approach in which

also depends on n and n .

For uniaxial tension, Ogden proposed

in the form

32 2

1

where

and

are real constants. This form allowed the author to fit the theory to the experimental data of

Holt and McPherson11

and consequently to account phenomenologically for the influence of

stress-induced crystallization on relative volume change.

To conclude on the modeling of the reversible change in volume of stretched rubber,

numerous expressions of the relative volume have been proposed. All of them predicted

volume change in the case of small deformations. However, for large deformations, the

predictions diverge. This is due to the fact that large deformations generate heterogeneity in

the bulk material, which is composed of numerous cavities which appear and grow. Finally, it

should be noted that for large deformations in crystallizable rubbers, the approaches of

Ogden73

and Sharda and Tschoegl75

account phenomenologically for the volume decrease due

to the crystallization of the stretched polymer chains.

C. NOTE ON MODELING THE IRREVERSIBLE CHANGE IN VOLUME

UNDER MULTIAXIAL LOADING CONDITIONS

Special experiments, as proposed by Gent and Lindley76

(1958), Gent and Wang77

(1991) and Legorju-Jago and Bathias78

(2002), were carried out to highlight the sudden

initiation and growth of cavities in bulk material. For modeling, the cavitation phenomenon

under hydrostatic loading conditions was studied considering the stability conditions for the

28

sudden growth of microscopic cavities in the incompressible bulk (see for instance

Ball79

(1982) and Horgan and Abeyaratne80

(1986)). This approach was then generalized to

the other loading conditions by Hou and Abeyaratne81

(1992). To take into account the

dependence of the stress-strain relationship on the growth of pre-existing cavities, the models

incorporate damage variables into compressible hyperelastic approaches (see the review by

Boyce and Arruda82

(2000)) to quantify the irreversible change in porosity.83-87

These models

can also be extended to cavitation by adapting the rate equation for the damage variable88

.

Nevertheless, they are limited to small porosity values, so that the growing cavities do not

intervene, and the irreversible change in volume seems to be associated with multiaxial

loading conditions. In fact, under uniaxial loading conditions, the fact that cavities initiate and

grow does not lead to a permanent set in terms of relative volume change.

V. CONCLUSIONS AND PERSPECTIVES

This paper has reviewed the literature on changes in the volume of rubber by gathering

observations reported from the end of the 19th

until now. Usually, the volume variation is

determined using dilatometry, but specific gravity, hydrostatic weighing or digital image

correlation can also be used. The fact that the measurement technique differs from one author

to another explains why the comparison of the results is difficult. Moreover, the strain rate,

the time used for measurement, the accuracy, etc, are rarely given. Again, the material

formulation is only precisely given in the recent studies. One can therefore imagine that there

are as many results as there are compositions, and in some cases as there are measurement

techniques. However, it seems reasonable to generalize the results obtained as follows:

(i) for non-crystallizable rubbers, the higher the elongation, the higher the relative volume

variation. When the rubbers are filled, the concavity of the curve obtained in terms of

29

relative volume change depends on the interaction between the fillers and the rubber matrix;

(ii) crystallizable rubbers behave like non-crystallizable rubbers up to the elongation at the

beginning of the crystallization. Above this elongation, the relative volume variation begins to

decrease. When fillers are added, they act on the one hand as deformation concentrators and

allow the crystallization to begin at a lower elongation, and on the other hand as amplifiers of

the cavitation and cavity growth phenomenon. This explains why no decrease in the relative

volume change is observed in filled crystallizable rubbers; only a decrease in the curve slope

is apparent. To conclude, the vulcanization system can also act as fillers do. For instance,

peroxide vulcanization, which links carbon atoms of macromolecules, concentrates the

deformation more, i.e. allows crystallization to begin at a lower elongation, than sulfur

vulcanization;

(iii) for both crystallizable and non-crystallizable filled rubbers under uniaxial cyclic loading,

volume variation is a reversible process during the first cycles. No residual change in volume

is observed after the first cycles. The maximum value of the relative volume change is

obtained during the first cycle. Contrary to stress, volume variation is stabilized after the first

cycle;

(iv) in crystallizable filled rubbers and contrary to non-crystallizable filled rubbers, a

hysteresis loop is also observed in the volume variation curve for the second and the third

cycles. This loop results from the difference between crystallization and crystallite melting

kinetics.

Concerning the modeling of the change in volume, all of the approaches proposed

predict volume change in the case of small deformations. For large deformations, however,

the predictions diverge. This is mainly due to the fact that these approaches do not account for

the change in the rubber microstructure.

30

As a perspective, volume variation in rubber stretched under multiaxial loading

conditions is a question of importance, and more particularly the fact that such loading

conditions could generate an irreversible volume change. Until now, the effect of loading

multiaxiality on volume variation has only been studied under the particular loading case of

hydrostatic tension.

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35

LISTE OF FIGURES

FIG. 1. - Change in volume of stretched rubbers at different elongations and with different

quantities of pigment (see Figures 1 to 7 in reference 9).

Fig. 2. - Change in volume of rubber on stretching to different elongations at 25°C (see Figure

6 in reference 11).

Fig. 3. - Change in volume of rubber when stretched and when released in a stepwise manner

at 25°C (see Figure 7 in reference 11).

FIG. 4. - Modeling of the volume variation in carbon black-filled butyl and styrene butadiene

rubber (see Figure 6 in reference 33).

FIG. 5. - A new mechanical quantity to analyze volume variation (see Figure 6 in reference

33).

FIG. 6. - Difference in the concavity of volume variation curves obtained for covering and

coupling agents (see Figures 5(a) and 5(a) in reference 37).

FIG. 7. - Relative volume variation in filled and unfilled natural rubber versus elongation (see

Figure 1(b) in reference 15).

FIG. 8. - Schematic illustration of the competition between decohesion/cavitation and

crystallization in terms of relative volume variation (see Figure 2 in reference 15).

FIG. 9. - Relative volume variation over the first mechanical cycle in natural rubber (NR).

FIG. 10. - Relative volume variation over the first mechanical cycle in NR. The maximum

elongation value is lower than that at the beginning of crystallization.

FIG. 11. - Relative volume variation over the first mechanical cycle in F-NR.

FIG. 12. - Relative volume variation over the third mechanical cycle in F-NR.

FIG. 13. - Relative volume variation over the third mechanical cycle in F-SBR.

36

0

20

40

60

80

100

120

140

0 200 400 600 800

increase in the

particle quantity

elongation to

fracture curves

rela

tive v

olu

me v

ari

ati

on

(%

)

elongation (%)

FIG. 1. - Change in volume of stretched rubbers at different elongations and with different

quantities of pigment (see Figures 1 to 7 in reference 9).

37

re

lative v

olu

me

variation (

%)

time (min)

100

99

98

0 3 6

450%

725%

elongation phase no elongation

Fig. 2. - Change in volume of rubber on stretching to different elongations at 25°C (see Figure

6 in reference 11).

38

98

99

100

0 100 200 300 400 500 600 700

elongation (%)

rela

tiv

e v

olu

me

va

ria

tio

n (

%)

Fig. 3. - Change in volume of rubber when stretched and when released in a stepwise manner

at 25°C (see Figure 7 in reference 11).

39

V

elongation

V

VI

VII

FIG. 4. - Modeling of the volume variation in carbon black-filled butyl and styrene butadiene

rubber (see Figure 6 in reference 33).

40

V

/

elongation

VI /

VII /

V /

FIG. 5. - A new mechanical quantity to analyze volume variation (see Figure 6 in reference

33).

41

rela

tive v

olu

me v

ari

ati

on

elongation

Silica fillers treated

by covering agent

Silica fillers treated

by coupling agent

FIG. 6. - Difference in the concavity of volume variation curves obtained for covering and

coupling agents (see Figures 5(a) and 5(a) in reference 37).

42

-1

0

1

2

3

4

5

1 2 3 4

cycle #1 filled

cycle #2 filled

cycle #3 filled

unfilled

rela

tiv

e v

olu

me

va

ria

tio

n (

%)

elongation

FIG. 7. - Relative volume variation in filled and unfilled natural rubber versus elongation

(see Figure 1(b) in reference 15).

43

rela

tive v

olu

me v

ari

ati

on

elongation

crystallization

decohesion/cavitation

Relative volume variation measured

FIG. 8. - Schematic illustration of the competition between decohesion/cavitation and

crystallization in terms of relative volume variation (see Figure 2 in reference 15).

44

rela

tive v

olu

me v

ari

ati

on

(%

)

elongation (%)

FIG. 9. - Relative volume variation over the first mechanical cycle in natural rubber (NR).

45

rela

tive v

olu

me v

ari

ati

on

(%

)

elongation (%)

FIG. 10. - Relative volume variation over the first mechanical cycle in NR. The maximum

elongation value is lower than that at the beginning of crystallization.

46

rela

tive v

olu

me v

ari

ati

on

(%

)

elongation (%)

FIG. 11. - Relative volume variation over the first mechanical cycle in F-NR.

47

rela

tive v

olu

me v

ari

ati

on

(%

)

elongation (%)

FIG. 12. - Relative volume variation over the third mechanical cycle in F-NR.

48

FIG. 13. - Relative volume variation over the third mechanical cycle in F-SBR.